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Rouché's theorem , named after  Eugène Rouché , states that for any two  complex -valued  functions   f  and  g   holomorphic  inside some region  {\displaystyle K}  with closed contour  {\displaystyle \partial K} , if  | g ( z )| < | f ( z )|  on  {\displaystyle \partial K} , then  f  and  f  +  g  have the same number of zeros inside  {\displaystyle K} , where each zero is counted as many times as its  multiplicity .  Assumption: This theorem assumes that the contour  {\displaystyle \partial K}  is simple, that is, without self-intersections. Please follow the links to know more: Link 1... Link 2... Link 3...

Those who are preparing for competitive examinations in mathematics may utilize... All the best...  1. Find the radius of convergence $\Sigma \frac{(-1)^n}{n}(z-i)^n$ 2. Find the limit $x log x$ as $x$ tend to 0. 3. Find the number of generators in $(Z_{60}, \oplus)$. 4. Let $J$ be the square matrix with all entries 1. Then the rank of $J$ is _____ 5. Let V be the space of all polynomials of degree n, with complex coefficients. Then the dim V over $\mathbb{R}$ is _____ 6. Give an example of a Lebesgue integrable but not Riemann integrable function. 7. If $Y=2X+3$, find the correlation coefficient between $X$ and $Y$. 8. Find the number of nonisomorphic abelian groups of order 60. 9. Let $G$ be a group of order 15. Then G is $A$. nonabelian $B$. Simple $C$. Cyclic $D$. None of these 10. Give an example of a group in which every subgroup is normal 11. In a cyclic group of order 36, no. Of subgroups of order 6 is _____ 12. In $(Z_{12}, \oplus)